An attempt to compare \(S_{\rm th}^{\rm MBCS}\)  and \(S_{\rm sp}^{\rm MBCS}\)  with the entropy calculated from the exact solution of the
PFM has been made in Fig. 6 of Ref. [1]} for the
\(N=10\) , \(\Omega =11\)  system.
First of all, the calculation of \(S_{\rm sp}^{\rm exact}\) 
(thick solid line in Fig. 6(a) of Ref. [1]}) is definitely
not correct.
The author of [1]} claims that \(S_{\rm sp}^{\rm exact}\) 
does not vanish at \(T=0\)  because occupation probabilities \(f_h < 1\) 
and \(f_p >0\) .
The problem with the third law of thermodynamics“at absolute
zero, any part of the body must be in a definite quantum state - namely
the ground state \(\ldots \)  the entropy of the body - the logarithm of its
statistical weight - is equal to zero” (see \(§\) 23, page 66 in [4]}).
in this calculation is caused by confusion of interacting particles with
noninteracting “quasiparticles”:
particle levels are not eigen states of the pairing Hamiltonian,
their occupation numbers do not obey Fermi-Dirac distribution
and because of that, they cannot be used in Eq. (REF ) which
represents the free Fermi-gas combinatorics.
Incorrectly calculated \(S_{\rm sp}^{\rm exact}\)  is also published in
Fig. 8 of [5]}.
